Misc 6
Three letters are dictated to three persons and an envelope is addressed to each of them, the letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.
Let the envelope be denoted by A, B, C
and the corresponding letters are a, b, c
The letters are inserted into the envelopes at random,
& each envelope contain exactly one letter
Possible combinations can be
Aa Bb Cc
Aa Bc Cb
Ab Ba Cc
Ab Bc Ca
Ac Bb Ca
Ac Ba Cb
So, Total number of possible cases = 6
∴ n(S) = 6
Let A be the event that at least one letter is in its proper envelope
A = {(Aa, Bb, Cc), (Aa, Bc, Cb), (Ac, Bb, Ca), (Ab, Ba, Cc)}
n(A) = 4
So,
P(A) = n(A)﷮n(S)﷯
= 4﷮6﷯
= 𝟐﷮𝟑﷯